2 IMPLICIT FUNCTIONS 273

Rn (since all derivatives of g at (XQ , j0) are equal to the corresponding deriva-
tives of/); hence, by (9.3.5.1), v maps a neighborhood of XQ in Rm into the
space Rn, and the uniqueness part of (10.2.1) therefore proves that the
restriction of v to W n Rm is identical to u. Q.E.D.

One of the most important applications of (10.2.1) is the following:

(10.2.5) Let E, F be two Banach spaces, f a continuously differentiable
mapping of a neighborhood V of x0 e E into F. If'ff(x0) is a linear homeo-
morphism ofE onto F, there exists an open neighborhood U c V ofx0 such that
the restriction off to U is a homeomorphism of U onto an open neighborhood
of y0 =zf(xo) in F. Furthermore, iffisp times continuously differentiable in
U (resp. analytic in U, E and F being finite dimensional), the inverse mapping
9 0//(U) onto U is p times continuously differentiable (resp. analytic) /«/(U).

Apply (10.2.1) to the function h(x, y) =/(x) — y, exchanging the roles of
x and y; as D^XQ , y0) =/'(^o)? we conclude that there is an open ball W
of center y0 in F and a continuous mapping # of W into E such that^(W) c U,
f(g(y)) = y in W and #(y0) = x0; furthermore, by (10.2.3) (resp. (10.2.4)), if
/is/? times continuously differentiable (resp. analytic),#is/? times continuously
differentiable (resp. analytic). From the identity f(g(y)) = y it follows that g
is injective in W, hence is a bijective continuous mapping of W onto
V = #(W) c U; moreover, #(W) =/~1(W) is open in E, and /is a homeo-
morphism of V = g(W) onto W, which ends the proof.

PROBLEMS

1. Let E, F be two Banach spaces, A an open neighborhood of a point JCQ e E,/a con-
tinuous mapping of A into F, which is differentiable at XQ (but not necessarily at other
points of A). Suppose /'(*(>) is a linear homeomorphism of E onto its image in F;
show that there is a neighborhood U <= A of x0 such that f(x) ^ f(x0) for every
xe\J such that jc ^ JCQ . (Observe that the assumption implies the existence of a
constant c> 0 such that \\f'(xQ) - s\\ ^ c\\s\\ for all s e E (5.5.1).)

2. Let /= (/i,/2) be the mapping of R2 into itself defined by fi(x^, x2) = xi; f2(xi, x2) =
x2 ~ x\ for x\ ^ x2, jf2(xi, x2)« (x2 - x\x2)lxl for 0^x2^ x\, and finally
/2(*i, — *a) = —fz(x\,X2) for x2 ^ 0. Show that / is differentiable at every point of
R2; at the point (0, 0), D/ is the identity mapping of R2 onto itself, but D/ is not
continuous. Show that in every neighborhood of (0,0), there are pairs of distinct points
*', x" such that/(xO = /(^0 (compare to (10.2.5)).2\foTxel9zl9z2 in H (compare to (10.1.1)); (2) for